Are there airports who still do curbside check-in these days (post-9/11)? Mostly, airlines like you to do online- and kiosk check-in instead of wait in line, but special needs cases are fine to do in-person wait-in-line check-in (just be sure to get to the airport extra early! I give myself two hours prior to flight time for domestic flights, since both waiting in line and going through security can take longer for me).
I travel with my service dog, so am not eligible to do online- or kiosk check-in. They are really supposed to charge me the first-bag fee of what the online fee is, but never do; not a huge deal, as I think it is only an extra two dollars or something. Sometimes getting my medical checked luggages (between one and three bags, depending on the type and length of the trip) through for free takes extra time, convincing, and showing a letter from the airline to me that those bags are indeed free. I got said letter after one airline agent rudely refused to check any of my medical bags (three) for free and charged me $350 or so (my non-medical bag was overweight, so that fee was valid and part of that cost)! When I complained to the airline, I asked for a response on letterhead so that I could use it in case the situation happened again. I have only had to pull out the letter once so far, but am glad I had it in that situation.
The last flight I was on (last week), I had one medical bag and the check-in guy made me open the bag so he could see that it was indeed medical-related supplies. A quick glance was all he needed; he didn't search the bag or go near it (he was at the computer and the bag was on the scale to the far side of my other bag). It was annoying, as I had to get the key out to unlock it, but I'm not going to complain about it because he did allow the bag for free (without me having to pull out my letter).
I check my medical bags for three reasons: one (the one I always use) is too large to be a carry-on, there are a lot of liquids in the contents of the bags (security takes long enough as it is without having to declare a bunch of liquids and explain to the agents who are unsure about them being allowed and all), and there is no way I can carry even the one extra bag (much less two or three bags) with me as carry-on.
I do keep meaning to make a tag for the bag with either a medical or handicapped symbol on it - this might help with check-in (likely not) but mostly hopefully the baggage handlers would see it and realize the bag gets priority on the plane so it doesn't get left behind. I am not sure if the bags' tags scanner info. reports the medical bags as such, so if they don't, the baggage guys have no idea the bag gets priority placement in the hold. Whether or not they will actually SEE the tag is left to chance, but it is better than nothing, right?
Anyway, if the check-in person tries to charge you for the medical bag (look up your airline's fees ahead of time so you know what is the normal baggage fee for your regular luggage, i.e. if the person tells you to pay $60 and you know the fee should only be $25 because the second bag is medical, you'll know to argue), tell them AGAIN that the certain bag is medical (lately they haven't been listening when I tell them, "1 regular bag, 1 medical bag," and take it to mean, "2 regular bags I can charge for,") and doesn't have a fee for it. If they flat-out refuse and cause a big issue, ask for a CRO (Complaint Resolution Officer/Official) and/or charge the fee to your card and then complain to the airline as soon as you can so they can refund you the money.
I don't think you can pay online for your bags during online check-in if you have a medical bag, as I was able to do online check-in once (during a contest promotion) and it gave me a message that I would have to do baggage check-in at the airport because of the medical bag (when I selected that I had a medical bag on the baggage page). Each airline is different, though.